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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Computing the $p$-Selmer group of an elliptic curve
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by Z. Djabri, Edward F. Schaefer and N. P. Smart PDF
Trans. Amer. Math. Soc. 352 (2000), 5583-5597 Request permission

Abstract:

In this paper we explain how to bound the $p$-Selmer group of an elliptic curve over a number field $K$. Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number fields of degree at most $p^2-1$. Our method is practical for $p=3$, but for larger values of $p$ it becomes impractical with current computing power. In the examples we have calculated, our method produces exactly the $p$-Selmer group of the curve, and so one can use the method to find the Mordell-Weil rank of the curve when the usual method of $2$-descent fails.
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Additional Information
  • Z. Djabri
  • Affiliation: Institute of Mathematics and Statistics, University of Kent at Canterbury, Canterbury, Kent, CT2 7NF, United Kingdom
  • Address at time of publication: Riskcare, Piercy House, 7 Copthall Avenue, London EC2R 7NJ, United Kingdom
  • Email: zmd1@ukc.ac.uk, zdjabri@riskcare.com
  • Edward F. Schaefer
  • Affiliation: Department of Mathematics, Santa Clara University, Santa Clara, California 95053
  • Email: eschaefe@math.scu.edu
  • N. P. Smart
  • Affiliation: Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol, BS12 6QZ, United Kingdom
  • Address at time of publication: Computer Science Department, Woodland Road, University of Bristol, Bristol, BS8 1UB, United Kingdom
  • Email: nsma@hplb.hpl.hp.com, nigel@cs.bris.ac.uk
  • Received by editor(s): October 28, 1998
  • Received by editor(s) in revised form: February 26, 1999, and March 17, 1999
  • Published electronically: August 21, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 5583-5597
  • MSC (2000): Primary 11G05, 11Y99; Secondary 14H52, 14Q05
  • DOI: https://doi.org/10.1090/S0002-9947-00-02535-6
  • MathSciNet review: 1694286